Contingency Tables and Tests of Association

A contingency table (also called a two-way table) displays the frequency distribution of two categorical variables. The chi-squared test of association determines whether the two variables are independent.

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Structure of a Contingency Table

An $r \times c$r×c contingency table has $r$r rows and $c$c columns:

	Column 1	Column 2	$\cdots$⋯	Column $c$c	Row Total
Row 1	$O_{11}$O11​	$O_{12}$O12​	$\cdots$⋯	$O_{1c}$O1c​	$R_1$R1​
Row 2	$O_{21}$O21​	$O_{22}$O22​	$\cdots$⋯	$O_{2c}$O2c​	$R_2$R2​
$\vdots$⋮	$\vdots$⋮	$\vdots$⋮		$\vdots$⋮	$\vdots$⋮
Row $r$r	$O_{r1}$Or1​	$O_{r2}$Or2​	$\cdots$⋯	$O_{rc}$Orc​	$R_r$Rr​
Col Total	$C_1$C1​	$C_2$C2​	$\cdots$⋯	$C_c$Cc​	$N$N

where $N$N is the grand total.

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Hypotheses

$H_0$H0​: There is no association between the two variables (they are independent).

$H_1$H1​: There is an association between the two variables (they are not independent).

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Expected Frequencies

Under $H_0$H0​ (independence), the expected frequency for cell $(i, j)$(i,j) is:

$E_{ij} = \frac{R_i \times C_j}{N}$Eij​=NRi​×Cj​​